On the convergence of the Bieberbach polynomials in regions with piecewise analytic boundary
Let G be a finite region in \({\mathbb{C}}\) with \(0\in G\), bounded by a piecewise analytic Jordan curve \(\Gamma\) without cusps, and let \(\lambda\pi\) \((0<\lambda <2)\) be the smallest exterior angle under which two analytic arcs of \(\Gamma\) meet. Let \(f_ 0\) denote the normalized conformal map from G onto \(\{\) w: \(| w| <r\}\). Then the Bieberbach polynomials \(\pi_ n\) of G with respect to \(z=0\) satisfy \[ \max \{| f_ 0(z)-\pi_ n(z)|:\;z\in \bar G\}=O(\log n)n^{- \gamma}\text{ with } \gamma =\lambda /(2-\lambda). \] The main idea in the proof is to estimate an \(L^ 2\)-norm \(\| f'_ 0-Q_ n\|_{L^ 2}\) by a maximum norm \(\| g-P_ n\|_{\infty}\) for \(g=f'_ 0P\), where \(P(z)=\Pi (z-z_ j)\) in which the \(z_ j\) are the corners of \(\Gamma\).
scientific article; zbMATH DE number 4192069